Method for enhancing the mixing of a compressible mixing layer

Numerical simulations have been done for a compressible mixing layer, in which the inflow speed on the low speed side was made to have periodic undulations, so as to see if this method could enhance the mixing effect of the layer. Systematic computations for a 2-D compressible mixing layer with Mach numberM _e = 0.6 have been done, and the results showed that the proposed method was indeed effective in enhancing the mixing.     

Existence of shocklets in a two-dimensional supersonic mixing layer and its influence on the flow structure

The spatial evolution of a T-S wave and its subharmonic wave, introduced at the inlet in a 2-D supersonic mixing layer, was investigated by using DNS. The relationship between the amplitude of the disturbance wave and the strength of the shocklet caused by the disturbance was investigated. We analyzed the shape of the disturbance velocity profile on both sides of the shocklet, and found that the existence of shocklet affected appreciably the disturbance velocity. The effects on the high speed side and low speed side of the mixing layer were found to be different     

Weak mixing implies mixing for maps on topological graphs

In section 1, we correct an error in the proof of Lemma 3.1 in L. Alsedà, M.A. del Río, and J.A. Rodríguez. Transitivity and dense periodicity for graph maps. J. Difference Equ. Appl. 9, 577–598, 2003. In section 2 we give a simple proof that weak mixing implies mixing for maps on topological graphs. The proofs can also be extended to (not necessarily compact) intervals, so in particular, this paper shows that for one-dimensional manifolds, weak mixing implies mixing.     

Fast mixing for independent sets,colorings, and other models on trees

We study the mixing time of the Glauber dynamics for general spin systems on the regular tree, including the Ising model, the hard‐core model (independent sets), and the antiferromagnetic Potts model at zero temperature (colorings). We generalize a framework, developed in our recent paper (Martinelli, Sinclair, and Weitz, Tech. Report UCB//CSD‐03‐1256, Dept. of EECS, UC Berkeley, July 2003) in the context of the Ising model, for establishing mixing time O(nlog n), which ties this property closely to phase transitions in the underlying model. We use this framework to obtain rapid mixing results for several models over a significantly wider range of parameter values than previously known, including situations in which the mixing time is strongly dependent on the boundary condition. We also discuss applications of our framework to reconstruction problems on trees. © 2006 Wiley Periodicals, Inc. Random Struct. Alg., 2007     

2-fold and 3-fold mixing: why 3-dot-type counterexamples are impossible in one dimension

V.A. Rohlin asked in 1949 whether 2-fold mixing implies 3-fold mixing for a stationary process (ξ_i )_i2ℤ, and the question remains open today. In 1978, F. Ledrappier exhibited a counterexample to the 2-fold mixing implies 3-fold mixing problem, the socalled 3-dot system, but in the context of stationary random fields indexed by ℤ². In this work, we first present an attempt to adapt Ledrappier's construction to the onedimensional case, which finally leads to a stationary process which is 2-fold but not 3-fold mixing conditionally to the σ-algebra generated by some factor process. Then, using arguments coming from the theory of joinings, we will give some strong obstacles proving that Ledrappier's counterexample can not be fully adapted to one-dimensional stationary processes.     

Fastest mixing Markov chain problem for the union of two cliques

A symmetric, random walk on a graph G can be defined by prescribing weights to the edges in such a way that for each vertex the sum of the weights of the edges incident to the vertex is at most one. The fastest mixing, Markov chain (FMMC) problem for G is to determine the weighting that yields the fastest mixing random walk. We solve the FMMC problem in the case that G is the union of two complete graphs.     

Direct numerical simulation of transition and turbulence in compressible mixing layer

The three-dimensional compressible Navier-Stokes equations are approximated by a fifth order upwind compact and a sixth order symmetrical compact difference relations combined with three-stage Ronge-Kutta method. The computed results are presented for convective Mach numberMc = 0.8 andRe = 200 with initial data which have equal and opposite oblique waves. From the computed results we can see the variation of coherent structures with time integration and full process of instability, formation of A -vortices, double horseshoe vortices and mushroom structures. The large structures break into small and smaller vortex structures. Finally, the movement of small structure becomes dominant, and flow field turns into turbulence. It is noted that production of small vortex structures is combined with turning of symmetrical structures to unsymmetrical ones. It is shown in the present computation that the flow field turns into turbulence directly from initial instability and there is not vortex pairing in process of transition. It means that for large convective Mach number the transition mechanism for compressible mixing layer differs from that in incompressible mixing layer.     

Rapid mixing of hypergraph independent sets

We prove that the mixing time of the Glauber dynamics for sampling independent sets on n‐vertex k‐uniform hypergraphs is when the maximum degree Δ satisfies Δ ≤ c2^k/2, improving on the previous bound Bordewich and co‐workers of Δ ≤ k ? 2. This result brings the algorithmic bound to within a constant factor of the hardness bound of Bezakova and co‐workers which showed that it is NP‐hard to approximately count independent sets on hypergraphs when Δ ≥ 5·2^k/2.     

Hereditarily hypercyclic operators and mixing

We show that hereditary transitivity (respectively strongly hereditary transitivity) is equivalent to weak mixing (respectively strong mixing) in a discrete dynamical system with Polish phase space. We also study the connection between local orbit structure and hypercyclicity, and obtain a “local hypercyclicity criterion.”     

Transitivity, mixing and chaos for a class of set-valued mappings

Consider the continuous map f : x → X and the continuous map f of K,(X) into itself induced by f, where X is a metric space and K(X) the space of all non-empty compact subsets of x endowed with the Hausdorff metric. According to the questions whether the chaoticity of f implies the chaoticity of f posed by Roman-Flores and when the chaoticity of f implies the chaoticity of f posed by Fedeli, we investigate the relations between f and f in the related dynamical properties such as transitivity, weakly mixing and mixing, etc. And by using the obtained results, we give the satisfied answers to Roman-Flores's question and Fedeli's question.     

Rapid mixing of Gibbs sampling on graphs that are sparse on average

Gibbs sampling also known as Glauber dynamics is a popular technique for sampling high dimensional distributions defined on graphs. Of special interest is the behavior of Gibbs sampling on the Erd?s‐Rényi random graph G(n,d/n), where each edge is chosen independently with probability d/n and d is fixed. While the average degree in G(n,d/n) is d(1 ‐ o(1)), it contains many nodes of degree of order log n/log log n. The existence of nodes of almost logarithmic degrees implies that for many natural distributions defined on G(n,p) such as uniform coloring (with a constant number of colors) or the Ising model at any fixed inverse temperature β, the mixing time of Gibbs sampling is at least n^{1+Ω(1/log log n)}. Recall that the Ising model with inverse temperature β defined on a graph G = (V,E) is the distribution over {±}^Vgiven by . High degree nodes pose a technical challenge in proving polynomial time mixing of the dynamics for many models including the Ising model and coloring. Almost all known sufficient conditions in terms of β or number of colors needed for rapid mixing of Gibbs samplers are stated in terms of the maximum degree of the underlying graph. In this work, we show that for every d < ∞ and the Ising model defined on G (n, d/n), there exists a β_d > 0, such that for all β < β_d with probability going to 1 as n →∞, the mixing time of the dynamics on G (n, d/n) is polynomial in n. Our results are the first polynomial time mixing results proven for a natural model on G (n, d/n) for d > 1 where the parameters of the model do not depend on n. They also provide a rare example where one can prove a polynomial time mixing of Gibbs sampler in a situation where the actual mixing time is slower than npolylog(n). Our proof exploits in novel ways the local tree like structure of Erd?s‐Rényi random graphs, comparison and block dynamics arguments and a recent result of Weitz. Our results extend to much more general families of graphs which are sparse in some average sense and to much more general interactions. In particular, they apply to any graph for which every vertex v of the graph has a neighborhood N(v) of radius O(log n) in which the induced sub‐graph is a tree union at most O(log n) edges and where for each simple path in N(v) the sum of the vertex degrees along the path is O(log n). Moreover, our result apply also in the case of arbitrary external fields and provide the first FPRAS for sampling the Ising distribution in this case. We finally present a non Markov Chain algorithm for sampling the distribution which is effective for a wider range of parameters. In particular, for G(n, d/n) it applies for all external fields and β < β_d, where d tanh(β_d) = 1 is the critical point for decay of correlation for the Ising model on G(n, d/n). © 2009 Wiley Periodicals, Inc. Random Struct. Alg., 2009     

Weakly coupled analysis of a blade in multiphase mixing vessel

Two or more physical systems frequently interact with each other, where the independent solution of one system is impossible without a simultaneous solution of the others. An obvious coupled system is that of a dynamic fluid-structure interaction. [8] In this paper a computational analysis of the fluid-structure interaction in a mixing vessel is presented. In mixing vessels the fluid can have a significant influence on the deformation of blades during mixing, depending on speed of mixing blades and fluid viscosity. For this purpose a computational weakly coupled analysis has been performed to determine the multiphase fluid influences on the mixing vessel structure. The multiphase fluid field in the mixing vessel was first analyzed with the computational fluid dynamics (CFD) code CFX. The results in the form of pressure were then applied to the blade model, which was the analysed with the structural code MSC.visualNastran forWindows, which is based on the finite element method (FEM). (© 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Spatial mixing and nonlocal Markov chains

We consider spin systems with nearest‐neighbor interactions on an n‐vertex d‐dimensional cube of the integer lattice graph . We study the effects that the strong spatial mixing condition (SSM) has on the rate of convergence to equilibrium of nonlocal Markov chains. We prove that when SSM holds, the relaxation time (i.e., the inverse spectral gap) of general block dynamics is O(r), where r is the number of blocks. As a second application of our technology, it is established that SSM implies an O(1) bound for the relaxation time of the Swendsen‐Wang dynamics for the ferromagnetic Ising and Potts models. We also prove that for monotone spin systems SSM implies that the mixing time of systematic scan dynamics is . Our proofs use a variety of techniques for the analysis of Markov chains including coupling, functional analysis and linear algebra.     

Tightness bounds for strongly mixing random sequences

For a given strictly stationary, strongly mixing random sequence for which the distributions of the partial sums are tight, certain ``tightness bounds" exist which depend only on the marginal distribution and the mixing rate.

     

The mixing time of Glauber dynamics for coloring regular trees

We consider Metropolis Glauber dynamics for sampling proper q‐colorings of the n‐vertex complete b‐ary tree when 3 ≤ q ≤ b/(2lnb). We give both upper and lower bounds on the mixing time. Our upper bound is n^{O(b/ log b)} and our lower bound is n^{Ω(b/(q log b))}, where the constants implicit in the O() and Ω() notation do not depend upon n, q or b. © 2010 Wiley Periodicals, Inc. Random Struct. Alg., 2010     

Torpid mixing of the Wang–Swendsen–Kotecký algorithm for sampling colorings

We study the problem of sampling uniformly at random from the set of k-colorings of a graph with maximum degree Δ. We focus attention on the Markov chain Monte Carlo method, particularly on a popular Markov chain for this problem, the Wang–Swendsen–Kotecký (WSK) algorithm. The second author recently proved that the WSK algorithm quickly converges to the desired distribution when k11Δ/6. We study how far these positive results can be extended in general. In this note we prove the first non-trivial results on when the WSK algorithm takes exponentially long to reach the stationary distribution and is thus called torpidly mixing. In particular, we show that the WSK algorithm is torpidly mixing on a family of bipartite graphs when 3k<Δ/(20logΔ), and on a family of planar graphs for any number of colors. We also give a family of graphs for which, despite their small chromatic number, the WSK algorithm is not ergodic when kΔ/2, provided k is larger than some absolute constant k₀.     

Quasi‐polynomial mixing of critical two‐dimensional random cluster models

We study the Glauber dynamics for the random cluster (FK) model on the torus with parameters (p,q), for q ∈ (1,4] and p the critical point p_c. The dynamics is believed to undergo a critical slowdown, with its continuous‐time mixing time transitioning from for p ≠ p_c to a power‐law in n at p = p_c. This was verified at p ≠ p_c by Blanca and Sinclair, whereas at the critical p = p_c, with the exception of the special integer points q = 2,3,4 (where the model corresponds to the Ising/Potts models) the best‐known upper bound on mixing was exponential in n. Here we prove an upper bound of at p = p_c for all q ∈ (1,4], where a key ingredient is bounding the number of nested long‐range crossings at criticality.     

Rank one lightly mixing

A rank one transformationT was constructed by Chacón that is weakly mixing but not mixing. We will show thatT is lightly mixing, not partially mixing, and not lightly 2-mixing. Partially supported by an NSF Postdoctoral Research Fellowship.     

Central limit theorem and the bootstrap for -statistics of strongly mixing data

The asymptotic normality of U-statistics has so far been proved for iid data and under various mixing conditions such as absolute regularity, but not for strong mixing. We use a coupling technique introduced in 1983 by Bradley [R.C. Bradley, Approximation theorems for strongly mixing random variables, Michigan Math. J. 30 (1983),69–81] to prove a new generalized covariance inequality similar to Yoshihara’s [K. Yoshihara, Limiting behavior of U-statistics for stationary, absolutely regular processes, Z. Wahrsch. Verw. Gebiete 35 (1976), 237–252]. It follows from the Hoeffding-decomposition and this inequality that U-statistics of strongly mixing observations converge to a normal limit if the kernel of the U-statistic fulfills some moment and continuity conditions.The validity of the bootstrap for U-statistics has until now only been established in the case of iid data (see [P.J. Bickel, D.A. Freedman, Some asymptotic theory for the bootstrap, Ann. Statist. 9 (1981), 1196–1217]. For mixing data, Politis and Romano [D.N. Politis, J.P. Romano, A circular block resampling procedure for stationary data, in: R. Lepage, L. Billard (Eds.), Exploring the Limits of Bootstrap, Wiley, New York, 1992, pp. 263–270] proposed the circular block bootstrap, which leads to a consistent estimation of the sample mean’s distribution. We extend these results to U-statistics of weakly dependent data and prove a CLT for the circular block bootstrap version of U-statistics under absolute regularity and strong mixing. We also calculate a rate of convergence for the bootstrap variance estimator of a U-statistic and give some simulation results.